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http://dx.doi.org/10.4134/JKMS.j210188

ON WEIGHTED COMPACTNESS OF COMMUTATORS OF BILINEAR FRACTIONAL MAXIMAL OPERATOR  

He, Qianjun (School of Applied Science Beijing Information Science and Technology University)
Zhang, Juan (School of Science Beijing Forestry University)
Publication Information
Journal of the Korean Mathematical Society / v.59, no.3, 2022 , pp. 495-517 More about this Journal
Abstract
Let Mα be a bilinear fractional maximal operator and BMα be a fractional maximal operator associated with the bilinear Hilbert transform. In this paper, the compactness on weighted Lebesgue spaces are considered for commutators of bilinear fractional maximal operators; these commutators include the fractional maximal linear commutators Mjα,β and BMjα,β (j = 1, 2), the fractional maximal iterated commutator ${\mathcal{M}}_{{\alpha},{\vec{b}}}$, and $BM_{{\alpha},{\vec{b}}}$, where b ∈ BMO(ℝd) and ${\vec{b}}\;=\;(b_1,b_2)\;{\in}\;BMO({\mathbb{R}}^d)\;{\times}\;BMO({\mathbb{R}}^d)$. In particular, we improve the well-known results to a larger scale for 1/2 < q < ∞ and give positive answers to the questions in [2].
Keywords
Bilinear fractional maximal operators; commutators; compactness; weighted esitmates;
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1 A. Benyi and R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3609-3621. https://doi.org/10.1090/S0002-9939-2013-11689-8   DOI
2 L. Chaffee and R. H. Torres, Characterization of compactness of the commutators of bilinear fractional integral operators, Potential Anal. 43 (2015), no. 3, 481-494. https://doi.org/10.1007/s11118-015-9481-6   DOI
3 Y. Chen and Y. Ding, Compactness of commutators of singular integrals with variable kernels, Chinese J. Contemp. Math. 30 (2009), no. 2, 153-166; translated from Chinese Ann. Math. Ser. A 30 (2009), no. 2, 201-212.
4 R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. https://doi.org/10.4064/sm-51-3-241-250   DOI
5 Q. He and P. Li, On weighted compactness of commutators of Schrodinger operators, Preprint, 2021, arXiv: 2102.01277.
6 T. Iida, A characterization of a multiple weights class, Tokyo J. Math. 35 (2012), no. 2, 375-383. https://doi.org/10.3836/tjm/1358951326   DOI
7 A. Benyi, W. Damian, K. Moen, and R. H. Torres, Compact bilinear commutators: the weighted case, Michigan Math. J. 64 (2015), no. 1, 39-51. https://doi.org/10.1307/mmj/1427203284   DOI
8 M. Cao, A. Olivo, and K. Yabuta, Extrapolation for multilinear compact operators and applications, Preprint, 2020, arXiv:2011.13191.
9 Y. Chen and Y. Ding, Compactness characterization of commutators for Littlewood-Paley operators, Kodai Math. J. 32 (2009), no. 2, 256-323. http://projecteuclid.org/euclid.kmj/1245982907   DOI
10 R. Bu and J. Chen, Compactness for the commutators of multilinear singular integral operators with non-smooth kernels, Appl. Math. J. Chinese Univ. Ser. B 34 (2019), no. 1, 55-75. https://doi.org/10.1007/s11766-019-3501-z   DOI
11 T. Iida, Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces, J. Inequal. Appl. 2016 (2016), Paper No. 4, 23 pp. https://doi.org/10.1186/s13660-015-0953-4   DOI
12 J. B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1985. https://doi.org/10.1007/978-1-4757-3828-5   DOI
13 W. Guo, Y. Wen, H. Wu, and D. Yang, On the compactness of oscillation and variation of commutators, Banach J. Math. Anal. 15 (2021), no. 2, Paper No. 37, 29 pp. https://doi.org/10.1007/s43037-021-00123-z   DOI
14 Q. He, M. Wei, and D. Yan, Weighted estimates for bilinear fractional integral operators and their commutators on Morrey spaces, Preprint, 2019, arXiv:1905.10946.
15 T. Iwaniec and C. Sbordone, Riesz transforms and elliptic PDEs with VMO coefficients, J. Anal. Math. 74 (1998), 183-212. https://doi.org/10.1007/BF02819450   DOI
16 S. S. Kutateladze, Fundamentals of Functional Analysis, Kluwer Texts in the Mathematical Sciences, 12, Kluwer Academic Publishers Group, Dordrecht, 1996. https://doi.org/10.1007/978-94-015-8755-6   DOI
17 A. Uchiyama, On the compactness of operators of Hankel type, Tohoku Math. J. (2) 30 (1978), no. 1, 163-171. https://doi.org/10.2748/tmj/1178230105   DOI
18 F. Beatrous and S.-Y. Li, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal. 111 (1993), no. 2, 350-379. https://doi.org/10.1006/jfan.1993.1017   DOI
19 X. Li, Q. He, and D. Yan, Weighted estimates for bilinear fractional integral operator of iterated product commutators on Morrey spaces, J. Math. Inequal. 14 (2020), no. 4, 1249-1267. https://doi.org/10.7153/jmi-2020-14-81   DOI
20 K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213-238. https://doi.org/10.1007/BF03191210   DOI
21 M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.
22 Q. Xue, Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators, Studia Math. 217 (2013), no. 2, 97-122. https://doi.org/10.4064/sm217-2-1   DOI
23 A. Benyi, W. Damian, K. Moen, and R. H. Torres, Compactness properties of commutators of bilinear fractional integrals, Math. Z. 280 (2015), no. 1-2, 569-582. https://doi.org/10.1007/s00209-015-1437-4   DOI
24 X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. https://doi.org/10.1016/j.jmaa.2009.08.022   DOI
25 S. G. Krantz and S.-Y. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications. I, J. Math. Anal. Appl. 258 (2001), no. 2, 629-641. https://doi.org/10.1006/jmaa.2000.7402   DOI
26 P. Li and L. Peng, Compact commutators of Riesz transforms associated to Schrodinger operator, Pure Appl. Math. Q. 8 (2012), no. 3, 713-739. https://doi.org/10.4310/PAMQ.2012.v8.n3.a7   DOI
27 H. Liu and L. Tang, Compactness for higher order commutators of oscillatory singular integral operators, Internat. J. Math. 20 (2009), no. 9, 1137-1146. https://doi.org/10.1142/S0129167X09005698   DOI
28 A. K. Lerner, S. Ombrosi, C. Perez, R. H. Torres, and R. Trujillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderon-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. https://doi.org/10.1016/j.aim.2008.10.014   DOI
29 Q. He and D. Yan, Bilinear fractional integral operators on Morrey spaces, Positivity 25 (2021), no. 2, 399-429. https://doi.org/10.1007/s11117-020-00763-9   DOI
30 B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. https://doi.org/10.2307/1996833   DOI
31 Q. Xue, K. Yabuta, and J. Yan, Weighted Frechet-Kolmogorov theorem and compactness of vector-valued multilinear operators, J. Geom. Anal. 31 (2021), no. 10, 9891-9914. https://doi.org/10.1007/s12220-021-00630-3   DOI
32 S. Wang and Q. Xue, On weighted compactness of commutators of bilinear maximal Calderon-Zygmund singular integral operators, Forum Math. 34 (2022), no. 2, 307-322. https://doi.org/10.1515/forum-2020-0357   DOI
33 D.-H. Wang, J. Zhou, and Z.-D. Teng, On the compactness of commutators of Hardy-Littlewood maximal operator, Anal. Math. 45 (2019), no. 3, 599-619. https://doi.org/10.1007/s10476-019-0818-z   DOI